Linear Dependence of Matrix Vectors

[zplot]

I need to prove that the set {I, A, A^2,..., A^n} is linear dependent where A is any nxn matrix. The vector space is the set of nxn matrix, considered as a nxn dimensional vector space.

Does anybody have an idea how to prove it?
Thank you very much.

 

[aspiring_one]

well I maybe wrong but to get you started:

I is in the nxn vector space and all the other matrices A, A^2, ..A^n

You might start by showing the linear combinations of matrices to get I and then relate that to the matrices A, of which matrices' can be written as nxn A^n=PA(^n)P^(-1)

 

[zplot]

Well, I tried to write A=P^(-1) J P where J is a Jordan matrix. Even more, I also tried to put J=N + D where N is nilpotent such that N^n=0 and D diagonal but I could not prove that the set is linear dependent. Thank you for your help. I f you have any further details or ideas I would be pleased.

 

[zplot]

At the end I arrived at the right solution. (A-lambda I)^k must be zero for some k<n. It comes from the Jordan canonical matrix, where lambda belongs to its spectrum. Logically then, the set {I, A, A^2,..., A^n} is linear dependent.

Thank you

 

[henry_m]

An alternative method, which gives you the exact dependence right off the bat, is to use the Cayley-Hamilton theorem. The characteristic polynomial is of degree at most n, and is satisfied by the matrix.

 

[zplot]

Great! Thats certainly a much better, simpler and more elegant solution.
Thank you very much, Henry.